Calculus Examples

$\ln^{5} (x)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\ln (x)$ .

$\frac{d}{d u} [u^{5}] \frac{d}{d x} [\ln (x)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$5 u^{4} \frac{d}{d x} [\ln (x)]$

Step 1.3

Replace all occurrences of $u$ with $\ln (x)$ .

$5 \ln^{4} (x) \frac{d}{d x} [\ln (x)]$

Step 2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$5 \ln^{4} (x) \frac{1}{x}$

Step 3

Combine fractions.

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Step 3.1

Combine $\frac{1}{x}$ and $5$ .

$\frac{5}{x} \ln^{4} (x)$

Step 3.2

Combine $\frac{5}{x}$ and $\ln^{4} (x)$ .

$\frac{5 \ln^{4} (x)}{x}$